Subsection [04X7]

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(2.6) Beware that, even though the subspace $\mathrm{Sk}(X)$ of $X^{\mathrm{an}}$ only depends on $X$ , the $\Delta$ -structure on $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$ and the retraction $\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X)$ depend on the choice of the (good) minimal dlt-model $\mathscr{X}$ ; we will illustrate this in Example 2.7 below. However, the essential skeleton $\mathrm{Sk}(X)$ does carry a canonical piecewise integral affine structure, which is induced by the embedding into the $K$ -analytic space $X^{\mathrm{an}}$ : see [MN15, §3.2]. If $\mathscr{X}$ is a minimal dlt-model for $X$ , then this piecewise integral affine structure coincides with the one induced by the $\Delta$ -complex structure on $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$ , provided that the barycentric coordinates on the faces of $\Delta(\mathscr{X}^{\mathrm{snc}})$ are weighted by the multiplicities of the prime components in $\mathscr{X}_{k}$ as in [MN15, 3.2.1].

Example 2.7.

Let $X$ be a maximally degenerate $K3$ surface over $K$ , and let $\mathscr{X}$ be a good minimal dlt-model over $R$ with reduced special fiber. Then $\mathrm{Sk}(X)$ is homeomorphic to $2$ -sphere, and $\Delta(\mathscr{X}^{\mathrm{snc}})$ provides this sphere with a triangulation. Different choice of $\mathscr{X}$ are related by elementary modifications (flops) of type 0, 1 or 2 [FM83, pp.12-15]. An elementary modification of type 0 does not affect the triangulation of $\mathrm{Sk}(X)$ or the map $\rho_{\mathscr{X}}$ , because it only changes $\mathscr{X}$ along a curve contained in a two-dimensional open stratum. An elementary modification of type 1 does not modify the triangulation of $\mathrm{Sk}(X)$ but it does alter the map $\rho_{\mathscr{X}}$ , because the points of $X^{\mathrm{an}}$ that specialize to the minus one curve that is flipped (but not to its intersection with a double curve) will be mapped to a different vertex of $\mathrm{Sk}(X)$ . Finally, an elementary modification of type 2 flips an edge in the triangulation of $\mathrm{Sk}(X)$ , but does not alter $\rho_{\mathscr{X}}$ because $\rho_{\mathscr{X}}$ is invariant under blow-ups of strata in snc-models (see Propositions 3.1.7 and 3.1.9 in [MN15]).

Proposition 2.8.

Let $X$ be a projective Calabi-Yau variety over $K$ . Then the essential skeleton $\mathrm{Sk}(X)$ is a strong deformation retract of $X^{\mathrm{an}}$ . If $\mathscr{X}$ is a good minimal dlt-model that satisfies the assumption in (2), then $\rho_{\mathscr{X}}$ is homotopic to the identity on $X^{\mathrm{an}}$ relative to $\mathrm{Sk}(X)$ .

Proof.

It is shown in [NX16a, 4.2.4] that $\mathrm{Sk}(X)$ is a strong deformation retract of $X^{\mathrm{an}}$ . This implies that every continuous retraction $X^{\mathrm{an}}\to\mathrm{Sk}(X)$ is homotopic to the identity on $X^{\mathrm{an}}$ relative to $\mathrm{Sk}(X)$ ; in particular, this is true for the retraction $\rho_{\mathscr{X}}$ . ∎